In the Riemannian case, the reformulation of Penrose's twistor construction was done by Atiyah, Hitchin and Singer in [1]}.
This formulation associates a natural almost-complex manifold to any real Riemannian 4-manifold \((M,g)\) .
The basic idea is to incorporate all almost-complex structures in \(M\)  into one
almost-complex structure in a larger manifold; in other words, to provide the space of almost-complex structures
itself with an almost-complex structure.
The space of complex structures at a point is \(\mathbb {CP}^{1}\)  (see Remark REF ),
so over the whole manifold one gets the projective spin bundle \(\mathbb {PS}\) .
The almost-complex structure of \(\mathbb {PS}\)  is described by Ward and Wells
[2]} as follows.
Since \(\mathbb {PS}\)  has a connection, at any point \(p=(x,\alpha )\in \mathbb {PS}\)  the tangent space
decomposes as \(T_{p}\mathbb {PS} = T_{x}M\oplus T_{\alpha }\mathbb {CP}^{1}\) .
On \(T_{x}M\)  we have the complex structure \(J=iK\)  defined by \(\alpha =[\alpha _{A}]\) , with \(K\)  given by (REF ),
and \(T_{\alpha }\mathbb {CP}^{1}\)  is the tangent space to a complex manifold so it naturally has a complex structure,
say \(J_{\mathbb {CP}^1}\) . Thus, the complex structure of \(T_{p}\mathbb {PS}\)  is \(\mathcal {J}={\rm diag}(J,J_{\mathbb {CP}^1})\) .
Atiyah et al. proved [1]} that \(\mathcal {J}\)  is integrable if and only
if the Weyl curvature of \(g\)  is (A)SD.
In that case, the space \(\mathbb {PS}\)  becomes a 3-dimensional complex manifold: this is the
twistor space of \(M\) . It coincides with the space of twistor surfaces in the complexification of \(M\) .
